Let $$T\left(q\right)=\sum _{k=1}^{\infty }d\left(k\right)q^k,\left|q\right|<1,$$ where $d\left(k\right)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we prove that $$H\left(q\right)=T\left(q\right)-\frac{log(1-q)}{log\left(q\right)}$$ is strictly increasing on $(0,1)$, while $$F\left(q\right)=\frac{1-q}{q}H\left(q\right)$$ is strictly decreasing on $(0,1)$. These results are then applied to obtain various inequalities, one of which states that the double inequality $$\alpha \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)}<T\left(q\right)<\beta \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)},0<q<1,$$ holds with the best possible constant factors $\alpha =\gamma$ and $\beta =1$. Here, $\gamma$ denotes Euler’s constant. This refines a result of Salem, who...

Let $\left[x\right]$ be an integer part of $x$ and $d\left(n\right)$ be the number of positive divisor of $n$. Inspired by some results of M. Jutila (1987), we prove that for $1<c<\frac{6}{5}$, $$\sum _{n\le x}d\left(\left[n^c\right]\right)=cxlogx+(2\gamma -c)x+O\left(\frac{x}{logx}\right),$$ where $\gamma$ is the Euler constant and $\left[n^c\right]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of ${\mathbb{Z}}_q$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $\left|A\right|\sim logq$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll logq$ and $logk\ll logq$, provided the...

We give some characterizations of the class ${}_m\left(\Omega \right)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{{log}^{\alpha }}^2$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{\alpha -1}}^2$, while if $\alpha >2$ and $0<\epsilon \le \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{{log}^{\alpha }}^2$ into $A_{{log}^{\alpha -2-\epsilon }}^2$.We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space ${ℬ}_{{log}^{\alpha }}$, $\alpha \in ℝ$, into itself, while $H$ maps ${ℬ}_{{log}^{\alpha }}$ into ${ℬ}_{{log}^{\alpha +1}}$.In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space ${ℬ}_{{log}^{\alpha }}^1$, $\alpha >0$, into ${ℬ}_{{log}^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps ${ℬ}_{{log}^{\alpha }}^1$, $\alpha \ge 0$,...

We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1-o\left(1\right))\alpha /\left(e^{\gamma }loglog(1/\beta )\right)$, which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $\mathbb{Z}{*}_m$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A,B\subseteq \mathbb{Z}{*}_m$ of densities α...

For any given positive integer k, and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

In this note, we show that the counting function of the number of composite positive integers $n\le x$ such that $\beta \left(n\right)={\sum }_{p\mid n}p$ is a prime is of order of magnitude at least $x/{(logx)}^3$ and at most $x/logx$.

In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that $\pi {\left(x\right)}^2<\frac{\mathrm{e}x}{logx}\pi \left(\frac{x}{\mathrm{e}}\right)$ for $x$ sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor $\frac{x}{logx}$ on its right hand side by the factor $\frac{x}{logx-h}$ for a given $h$, and by replacing the numerical factor $\mathrm{e}$ by a given positive $\alpha$. Finally, we introduce and study inequalities...

For the real quadratic map $P_a\left(x\right)=x^2+a$ and a given $ϵ\&gt;0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_a^n{|}_J$ univalent, with bounded distortion and $B(0,ϵ)\subseteq P_a^n\left(J\right)$ for some $n\in \mathbb{N}$. The $ϵ$-weakly expanding set is the set of points which do not have good expansion properties. Let $\alpha$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[\alpha ,-\alpha ]$. We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff...

For any positive integer k and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all n ≥ n₀, we have $r_{1,k}(A,n)\to \infty$ as n → ∞.

We prove that for i ≥ 1, the arithmetic $I\Delta ₀+{\Omega }_i$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1+\epsilon )lo{g}^{i+2}$, where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $lo{g}^{i+3}$.

Let K be an algebraic number field with non-trivial class group G and ${}_K$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_k\left(x\right)$ denote the number of non-zero principal ideals $a_K$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_k\left(x\right)$ behaves for x → ∞ asymptotically like $x{\left(logx\right)}^{1-1/\left|G\right|}{\left(loglogx\right)}^{{}_k\left(G\right)}$. We prove, among other results, that $₁\left(C_{n₁}\oplus C_{n₂}\right)=n₁+n₂$ for all integers n₁,n₂ with 1 < n₁|n₂.

Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_A/f({\partial }_A)$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda \to \log \max g(f^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g(f^{-1}(\lambda ))$ è analitica nel senso di Oka.

Let $f:I\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ for $t$ close to $1$, and also that the pressure function $t\mapsto P\left({\phi }_t\right)$ is analytic on an appropriate interval near $t=1$.

Let ${𝒯}_n$ denote the set of log canonical thresholds of pairs $(X,Y)$, with $X$ a nonsingular variety of dimension $n$, and $Y$ a nonempty closed subscheme of $X$. Using non-standard methods, we show that every limit of a decreasing sequence in ${𝒯}_n$ lies in ${𝒯}_{n-1}$, proving in this setting a conjecture of Kollár. We also show that ${𝒯}_n$ is closed in $𝐑$; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in...

The inequality $ʃlog|\sum a_n{e}^{2\pi i{\phi }_n}|d{\phi }_1\dots d{\phi }_n\ge Clog\left(\sum \right|a_n{{|}^2)}^{1/2}$ with an absolute constant C, and similar ones, are extended to the case of $a_n$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2\pi i\phi }$.